Infinite-server queues with batch arrivals and dependent service times (Q2909824)

This paper considers the heavy-traffic (HT) stochastic process limits for the queue length process (the number of customers in a system) in an infinity server (IS) \(\text{G}_{t}\)/\(\text{G}^{D}\)/\(\infty\) queueing model, having a general arrival process with time-varying arrival rate (\(\text{G}_{t}\)) and weakly dependent service time. Functional central limit theorems (FCLTs) were used for the sequential empirical processes driven by dependent service times. From the HT limits, it has been observed that the dependence among service times does not affect the mean queue length, but can significantly affect queue length variance. However, the derived variance formula takes a complicated form depending on the joint bivariate distribution of each pair of service times.NEWLINENEWLINEMoreover, the new G/G/IS batch model is introduced as a special case of the \(\text{G}_{t}\)/\(\text{G}^{D}\)/\(\infty\) model mentioned. For this IS batch model, new performance measures such as the arrival process of batches, the batch-size distribution, the service distribution for each customer and the dependence assumed for the service times of the customers in the same batch are presented.NEWLINENEWLINEAn explicit expression for the HT approximation of the variance function in terms of the mean values of the minimum of two independent and dependent service times and their associated stationary excess (or equilibrium residual lifetime) distribution is derived for this batch model with time varying arrival rates. Additionally, approximations with a sinusoidal arrival rate are provided for all the HTs.NEWLINENEWLINEIn sum, this paper presents progress made in the study of infinite server queues with batch arrivals and dependent service times.